The preface to a textbook frequently contains the author's justification for offering the public "another book" on the given subject. For our chosen topic, the arithmetic of elliptic curves, there is little need for such an apologia. Considering the vast amount of research currently being done in this area, the paucity of introductory texts is somewhat surprising. Parts of the theory are contained in various books of Lang (especially [La 3] and [La 5]); and there are books of Koblitz ([Kob]) and Robert ([Rob], now out of print) which concentrate mostly on the analytic and modular theory. In addition, survey articles have been written by Cassels ([Ca 7], really a short book) and Tate ([Ta 5J, which is beautifully written, but includes no proofs). Thus the author hopes that this volume will fill a real need, both for the serious student who wishes to learn the basic facts about the arithmetic of elliptic curves; and for the research mathematician who needs a reference source for those same basic facts. Our approach is more algebraic than that taken in, say, [La 3] or [La 5], where many of the basic theorems are derived using complex analytic methods and the Lefschetz principle. For this reason, we have had to rely somewhat more on techniques from algebraic geometry. However, the geomยญ etry of (smooth) curves, which is essentially all that we use, does not require a great deal of machinery
CONTENT
I Algebraic Varieties -- II Algebraic Curves -- III The Geometry of Elliptic Curves -- IV The Formal Group of an Elliptic Curve -- V Elliptic Curves over Finite Fields -- VIElliptic Curves over ? -- VII Elliptic Curves over Local Fields -- VIII Elliptic Curves over Global Fields -- IX Integral Points on Elliptic Curves -- X Computing the Mordell-Weil Group -- Appendix A Elliptic Curves in Characteristics 2 and 3 -- ยง1. Cohomology of Finite Groups -- ยง2. Galois Cohomology -- ยง3. Non-Abelian Cohomology -- Appendix C Further Topics: An Overview -- ยง11. Complex Multiplication -- ยง12. Modular Functions -- ยง13. Modular Curves -- ยง14. Tate Curves -- ยง15. Nรฉron Models and Tateโ{128}{153}s Algorithm -- ยง17. Duality Theory -- ยง18. Local Height Functions -- ยง19. The Image of Galois -- ยง20. Function Fields and Specialization Theorems -- Notes on Exercises -- List of Notation
